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Factorial Formula Calculator

Factorial Formula:

\[ n! = \prod_{k=1}^n k = 1 \times 2 \times 3 \times \cdots \times n \]

(0-170)

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1. What is Factorial?

The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. By definition, 0! = 1. Factorials grow very rapidly with increasing n.

2. How Does the Calculator Work?

The calculator uses the factorial definition:

\[ n! = \prod_{k=1}^n k = 1 \times 2 \times 3 \times \cdots \times n \]

Where:

Explanation: The calculator computes the product of all positive integers from 1 to n. For n=0, it returns 1 by definition.

3. Applications of Factorial

Details: Factorials are fundamental in combinatorics, probability theory, algebra, and mathematical analysis. They appear in permutations, combinations, Taylor series, and more.

4. Using the Calculator

Tips: Enter any integer between 0 and 170. The limit is 170 because 171! exceeds PHP's maximum float size. Results are formatted with thousand separators for readability.

5. Frequently Asked Questions (FAQ)

Q1: Why is 0! equal to 1?
A: This is by definition to maintain consistency in combinatorial formulas and the empty product concept.

Q2: Can factorials be calculated for negative numbers?
A: No, standard factorial is only defined for non-negative integers. For negative numbers, see Gamma function.

Q3: Why is there a limit of 170?
A: 171! ≈ 1.24×10³⁰⁹ exceeds PHP's floating point maximum. Higher values would require arbitrary-precision arithmetic.

Q4: How are factorials used in probability?
A: They're essential in permutation and combination calculations which form the basis of many probability problems.

Q5: What's the fastest way to compute large factorials?
A: For very large n, approximation methods like Stirling's formula are used: \( n! \approx \sqrt{2\pi n}(\frac{n}{e})^n \).

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